Multiple-Input Multiple-Output (MIMO) technology is a technology capable of improving data transmission/reception efficiency using multiple transmit (Tx) antennas and multiple receive (Rx) antennas instead of using a single Tx antenna and a single Rx antenna. In particular, a transmitting end or a receiving end of a wireless communication system can increase capability or improve performance using multiple antennas. Accordingly, the MIMO technology can also be referred to as a multi-antenna technology.
In order to support multi-antenna transmission, it may be able to apply a precoding matrix to appropriately distribute transmission information to each antenna according to a channel status and the like. A legacy 3GPP (3rd Generation Partnership Project) LTE (Long Term Evolution) system supports maximum 4 transmission antennas (4 Tx) to perform downlink transmission and defines a precoding codebook according to the transmission antennas.
In a multi-antenna system-based cellular communication environment, data transfer rate can be enhanced via beamforming between a transmitting end and a receiving end. Whether to apply a beamforming scheme is managed based on channel information. In general, it may be able to use a scheme that a receiving end appropriately quantizes a channel estimated by a reference signal and the like using a codebook and gives a transmitting end feedback on the quantized channel.
In the following, a spatial channel matrix (simply, channel matrix) capable of being used for generating a codebook is briefly explained. The spatial channel matrix (or, channel matrix) can be represented as follows.
      H    ⁢          (              i        ,        k            )        =      [                                                    ⁢                                          h                                  1                  ,                  1                                            ⁡                              (                                  i                  ,                  k                                )                                                                                        h                              1                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              1                ,                Nt                                      ⁡                          (                              i                ,                k                            )                                                                                      h                              2                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              2                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              2                ,                Nt                                      ⁡                          (                              i                ,                k                            )                                                            ⋮                          ⋮                          ⋱                          ⋮                                                                h                              Nr                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              Nr                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              Nr                ,                Nt                                      ⁡                          (                              i                ,                k                            )                                            ]  
In this case, H (i, k) corresponds to a spatial channel matrix, Nr corresponds to the number of reception antennas, Nt corresponds to the number of transmission antennas, r corresponds to an index of an reception antenna, t corresponds to an index of a transmission antenna, i corresponds to an index of an OFDM (or SC-FDMA) symbol, and k corresponds to an index of a subcarrier.
hr,t (i, k) corresponds to an element of a channel matrix H (i, k) indicating a state of an rth channel and a tth antenna on an ith symbol and kth subcarrier.
A spatial channel covariance matrix capable of being used in the present invention is briefly explained in the following. The spatial channel covariance matrix can be represented by such a sign as R. In particular, the spatial channel covariance matrix can be represented as
  R  =            E      ⁡              [                              H                          i              ,              k                        H                    ⁢                      H                          i              ,              k                                      ]              .  In this case, H and R correspond to a spatial channel matrix and a spatial channel covariance matrix, respectively. E[ ] corresponds to a mean, i corresponds to a symbol index, and k corresponds to a frequency index.
SVD (singular value decomposition) is one of important methods for decomposing a rectangular matrix. The SCD is widely used in signal processing and statistics. The SVD generalizes a spectrum theory of a matrix in response to a random rectangular matrix. An orthogonal square matrix can be decomposed to a diagonal matrix using the spectrum theory based on an Eigen value. Assume that a channel matrix H corresponds to m x n matrix consisting of a set element of real numbers or complex numbers. In this case, the matrix H can be represented by multiplication of three matrixes described in the following.Hm×n=Um×mΣm×nVn×nH 
In this case, U and V correspond to unitary matrixes and Σcorresponds to m×n diagonal matrix including a singular value which is not a negative value. The singular value corresponds to Σ=diag(σ1 . . . σr), σi=√{square root over (λi)}. As mentioned above, when a matrix is represented by multiplication of three matrixes, it is referred to as singular value decomposition. It may be able to handle a much more general matrix using the singular value decomposition compared to Eigen value decomposition capable of decomposing an orthogonal squre matrix only. The singular value decomposition and the Eigen value decomposition are related to each other.
When a matrix H corresponds to an Hermite matrix which is positive definite, all Eigen values of the H correspond to real numbers which are not negative numbers. In this case, a singular value and a singular vector of the H correspond to real numbers which are not negative numbers. In particular, the singular value and the singular vector of the H become identical to the Eigen value and the Eigen vector of the H. Meanwhile, EVD (Eigen value Decomposition) can be represented as follows (in this case, Eigen value may correspond to λ1, . . . , λr).HHH=(UΣVH)(UΣVH)H=UΣΣTUH HHH=(UΣVH)H(UΣVH)H=VΣTΣV 
In this case, Eigen value may correspond to λ1, . . . , λr. When singular value decomposition is performed on HHH, it is able to know information on U among U and V that indicate channel direction. When singular value decomposition is performed on HHH , it is able to know information on V. In general, each of a transmitting end and a receiving end performs beamforming to achieve a higher transfer rate in MU-MIMO (multi user-MIMO). If a beam of the receiving end and a beam of the transmitting end are represented by a matrix T and a matrix W, respectively, a channel to which beamforming is applied can be represented as THW=TU(Σ)VW. Hence, it may be preferable to generate a reception beam on the basis of the U and generate a transmission beam on the basis of the V to achieve a higher transfer rate.
In general, main concern in designing a codebook is to reduce feedback overhead using the number of bits as small as possible and precisely quantify a channel to achieve sufficient beamforming gain. One of schemes of designing a codebook, which is proposed or selected by recent communication standard such as 3GPP LTE (3rd Generation Partnership Project Long Term Evolution), LTE-Advanced, IEEE 16m system, etc. corresponding to an example of a mobile communication system, is to transform a codebook using a long-term covariance matrix of a channel as shown in equation 1 in the following.Wt=norm(RW)   [Equation 1]
In this case, W corresponds to a legacy codebook for reflecting short-term channel information, R corresponds to a long-term covariance matrix of a channel H, and norm (A) corresponds to a normalized matrix that norm is normalized by 1 according to each column of a matrix A. W′ corresponds to a final codebook transformed from the legacy codebook W using the channel matrix H, the long-term covariance matrix R of the channel matrix H and a norm function.
The R, which is the long-term covariance matrix of the channel matrix H, can be represented as equation 2 in the following.
                    R        =                              E            ⁡                          [                                                H                  H                                ⁢                H                            ]                                =                                    VΛV              H                        =                                          ∑                                  i                  =                  1                                Nt                            ⁢                                                          ⁢                                                σ                  i                                ⁢                                  v                  i                                ⁢                                                      v                    i                                    H                                                                                        [                  Equation          ⁢                                          ⁢          2                ]            
In this case, if the singular value decomposition is performed on the R, which is the long-term covariance matrix of the channel matrix H, the R is decomposed to VΛVH. V corresponds to Nt×Nt unitary matrix and has Vi as an ith column vector. Λ corresponds to a diagonal matrix and has σi as an ith diagonal component. VH corresponds to an Hermitian matrix of the V. And, σi, vi respectively correspond to an ith singular value and an ith singular column vector corresponding to the ith singular value (σ1≧σ2≧ . . . ≧σNt).